Both DeChambeau and Rahm finished on 26 under-par over 72 holes, but the American claimed his fifth LIV Golf title on the first extra hole with a birdie.
A tearful DeChambeau revealed he has faced challenges off the course this week without elaborating.
"A lot had happened in my life in the last week, I am so grateful for my team and everybody supporting me," he said.
"Golf is a fickle game, you work so hard at it your whole life, and then you realise golf is just golf."
DeChambeau’s quartet, the Crushers, which also includes Paul Casey, Charles Howell III and Anirban Lahiri, won the team event as they beat South Africa’s Southern Guards by a single shot with 76 under-par.
Officials on Sunday announced the event in South Africa would be staged again in 2027, though with revised dates of April 22-25.
"South Africa was unbelievable, it has to be the best LIV Golf event we have ever had," DeChambeau said.
More than 100,000 spectators were expected over the four days in Johannesburg.